New Results On The Single Server Queue With a Batch Markovian Arrival Process

ABSTRACT

The versatile Markovian point process was introduced by M. F. Neuts in 1979. This is a rich class of
point processes which contains many familiar arrival process as very special cases. Recently, the Batch
Markovian Arrival Process, a class of point processes which was subsequently shown to be equivalent to
Neuts' point process, has been studied using a more transparent notation.

Recent results in the matrix-analytic approach to queueing theory have substantially reduced the
computational complexity of the algorithmic solution of single server queues with a general Markovian
arrival process. We generalize these results to the single server queue with the batch arrival process and
emphasize the resulting simplifications.

Algorithms for the special cases of the PH/G/1 and MMPP/G/1 queues are highlighted as these
models are receiving renewed attention in the literature and the new algorithms proposed here are simpler
than existing ones. In particular, the PH/G/I queue has additional structure which further enhances the
efficiency of its algorithmic solution. Also, the two-state MMPP/G/1 queue, which has applications in
communications modeling, has an extremely simple solution.

1. INTRODUCTION

The versatile Markovian point process was introduced by M. F. Neuts in [1]. This is a very rich class
of point processes which contains many well known arrival processes as special cases. Among them are
the phase-type (PH) renewal process, the Markov modulated Poisson process (MMPP), overflows from
finite Markovian queues, etc. In each case, arrivals are allowed to occur in batches where different types
of arrivals can have different batch size distributions. The price paid for such generality was an elaborate
notation required to keep track of the different types of arrivals. Although the notation was complex, the
analysis of queues with this point process as the arrival stream proceeded, conceptually, in an analogous
fashion to that of queues with simpler arrival streams. Thus it was possible to solve in a unified
methodical analysis a whole class of queueing problems, unifying many results in the literature.

This was first accomplished by V. Ramaswami for the single server queue with the versatile
Markovian point process as the arrival stream [2]. Since then, the infinite server, c-server (with
deterministic service times), and finite queue versions have been solved, see [3], [4], and [5]. Although
the computational algorithm suggested by Rarnaswami's analysis has been shown to be numerically
stable [6], in practice it has not been feasible to implement it in its full generality. The setup computations
alone are a formidable burden on both CPU time and storage. Thus, until now, practical numerical
solutions have been limited to particular cases of the general model.

In our analysis of a single server queue with server vacations [7], we desired the solution to the queue
with a PH-renewal arrival process and the one with a correlated arrival stream such as an MMPP. As our
focus was not on batch arrivals, we did not proceed with the full generality of the versatile Markovian
point process, but constructed a new process which contained both PH-renewal and the MMPP processes
yet whose notation was very simple. We called this process the Markovian Arrival Process (MAP). This
construction is easily generalized to the Batch Markovian Arrival Process (BMAP) to allow for batch
arrivals. Although this new class of processes was originally thought to be more general than the versatile
Markovian point process, we later showed that the two processes were in fact equivalent. The only
difference is that the BMAP involves much simpler notation.

Special cases of the BMAP/G/I queue have received renewed attention in the communications
modeling literature. The interrupted Poisson process has long been used to approximate the overflow
traffic of finite trunk systems [8]. More recently, modeling of packetized voice and data traffic has
required consideration of more complicated arrival processes than the Poisson process. It is now well
known ([9], [10]) that the interarrival times in the packet streams are strongly correlated. The MMPP was
used in [10] to approximate the superposition of packetized voice processes and in [11] for a related
process. The MMPP was chosen because it is a tractable, non-renewal stream which could match certain
statistical properties of the original traffic. The MMPP/G/l queue approximated the first two moments of
delays as well as the tail probabilities with high accuracy. Other algorithms for solving the MMPP/G/l
queue are presented in [12] and [13]. For a case where the MMPP is obtained as the superposition of interrupted Poisson processes see [14]. Other special cases of the BMAP/G/I queue which have appeared
in the literature are related to the PH/G/I queue. We refer to the extended, annotated bibliography [15]
for many examples and special cases.

We present here new resnlts for the BMAP/G/I queue. In particular, we show that the matrix G,
which arises in the matrix analytic approach to queues of M/G/I type and is the key ingredient to the
computational procedures, has an exponential form. This exponential fonn leads to an efficient algorithm
for the computation of G as well as the coefficient matrices in the transition probability matrix of the
Markov chain embedded at departures. These are needed to compute the queue length distribution at
departures and at arbitrary times. This key result generalizes similar results in [7] ,and [16]. The
algorithms presented here allow for a general implementation of canned computer programs for solving
the general model. Such a program could be used for comparing vastly different arrival processes
entering a single server queue.

A further use of this algorithm is to evaluate the performance of superpositions of renewal processes
entering a queue. If the renewal processes are of phase type then the superposition is a special case of the
BMAP. Although the size of the matrices involved grows geometrically as the number of streams, for two
or three streams the computations are completely feasible. The delay seen by customers in the individual
streams can be derived from the results presented earlier. Similar calculations for the MMPP/M/c/c +K
queue were presented in [17]. These exact expressions could be used to validate various simple
approximations that have been proposed, see e.g., [18] and [19].

The remainder of this paper is organized as follows. In Section 2, we define the BMAP and present
some familiar special cases of the process. Section 3 consists of an outline of the traditional matrixanalytic
approach to solving the single server queue with a BMAP as the arrival stream emphasizing the
framework of the new notation. New results for the BMAP/G/I queue are presented in Section 4. Section
5 summarizes the algorithmic simplifications for the general model, highlighting the substantial savings in
both computational complexity and storage which are afforded by the new results. In Section 6 present
several special cases which have particularly simple solutions. Conclusions are presented in Section 7.